ewea 2011 brussels, belgium: europe’s premier wind energy event

EWEA 2011 Brussels, Belgium:Europe’s Premier Wind Energy Event

EWEA 2011 Brussels, Belgium:Europe’s Premier Wind Energy Event

Structural reliability analysis of rotor blades in ultimate loading

K. C. Bacharoudis1, D. J. Lekou2, T. P. Philippidis1 1. University of Patras, Dept. of Mechanical Engng & Aeronautics, Greece2. Centre for Renewable Energy Sources, Wind Energy Division, Greece

Brussels, 14-17 March 2011

The objectives are:

• Reliability assessment of a given design (safety factors, Target reliability)

• New probabilistic design, sensitivity studies (Improve/optimize blade structural design)

Probabilistic Strength AnalysisProbabilistic Strength Analysis


• Sectional analysis based on thin wall multi cellular theory was performed by a numerical tool (THIN)

• Monte Carlo, Edgeworth expansion method and Response Surface method/Monte Carlo were used

• Stress resultants, engineering elastic constants (E1, E2 , v12 , G12) and failure stresses (XT, XC, YT, YC, S) in the principal coordinate system of the UD ply were considered random variables (RV)



• Stochastic representation of material properties• Stochastic modeling of the stress resultants acting on

the blade section• Implementation of fast and accurate reliability

methods

The effort consists of three major tasks:



Sec 9.2m

Thin model

Blade model

Blade section

Rotor Blade ModelRotor Blade Model


Laminate

Material Property

Mean Std. Rd

Ex (GPa) 22.9 2.1 22.9

Ey (GPa) 7.9 1.6 7.9

vxy0.3 0.05 0.3

Gxy (GPa) 1.7 0.4 1.7

XT (Mpa) 241.2 34.4 134.2

XC (Mpa) 199.5 19.9 123.6

YT (Mpa) 22.0 3.6 11.5

YC (Mpa) 89.3 9.1 55.1

S (Mpa) 9.7 1.4 5.4

Material properties were considered normally distributed

Material properties at the ply level

Stochastic Material PropertiesStochastic Material Properties

Sec 9.2m

THIN model

Blade section


z

Nz

Mz

Nx

xMx

NyMy

y

Extreme Load Analysis (IEC-61400 ed.3)Extreme Load Analysis (IEC-61400 ed.3)

Directions of the stress resultants of the blade under extreme loading


0

200

400

600

800

1,000

1,200

1,400

1,600

1,800

2,000

1000 1200 1400 1600 1800 2000

Flap

mom

ent [

kN*m

]

Sample #

Time series from aero elastic codes

z

Nz

Mz

Nx

xMx

NyMy

y



0

200

400

600

800

1,000

1,200

1,400

1,600

1,800

2,000

1000 1200 1400 1600 1800 2000

Flap

mom

ent [

kN*m

]

Sample #

Sample valuesThresholdMax

z

Nz

Mz

Nx

xMx

NyMy

y


Pick local maxima according IEC 61400-1 Ed.3

Thrs.=mv+1.4*sd


-2

0

2

4

6

8

10

12

14

1300 1500 1700 1900 2100

-Ln(

-Ln(

F max

))

Flap moment [kN*m]

EmpiricalLognormalWeibull3pWeibullGumbelGumbel_1

Find best-fit probability distributions for every mean wind speed bin

z

Nz

Mz

Nx

xMx

NyMy

y



1.00E-12

1.00E-11

1.00E-10

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

300 1300 2300 3300

P(F e

xt>=

F/T=

10 m

in)

Flap moment [kN*m]

Lognormal

Weibull

3pWeibull

Gumbel

Gumbel_1

P=3.8e-7

Maximum observation

Pr , Pr ,ob F F T P F T ob F F V T p V dVext e extV

V

in

out

z

Nz

Mz

Nx

xMx

NyMy

y



0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

2730 2830 2930 3030 3130 3230 3330

P(F e

xt<F

/T=2

0 yr

s)

Flap moment [kN*m]

EmpiricalNormalLognormalWeibull3pWeibullGumbel

P F F T yr P F F Text ext

N

20 1 10min

z

Nz

Mz

Nx

xMx

NyMy

y



Stress resultants are modeled as Gumbel distribution

F x e e

x b

a

z

Nz

Mz

Nx

xMx

NyMy

y


Section 9.2mFd a b

Nx [kN] 306.00 1.297 243.6Ny [kN] 200.50 5.163 155.6Nz [kN] 145.87 4.289 112.7

Mx [kNm] 17.80 0.667 13.6Mz [kNm] 1872.50 50.610 1451.3My [kNm] 1253.75 35.510 970.4


0

0.2

0.4

0.6

0.8

1

1 6 11 16 21 26 31 36

Corr

elati

on

Sample #

Edge M.-TorsionEdge M.-Flap M.Edge M.-Flap F.Edge M.-Edge M.Edge M.-Edge F.Correlation matrix of stress resultants for section

9.2m from root

Tor. Flap M. Flap F Edge M. Edge F.

Tor. 1 0.48 0.53 0.77 0.79

Flap M. 0.48 1 0.95 0.33 0.32

Flap F. 0.53 0.95 1 0.28 0.27

Edge M. 0.77 0.33 0.28 1 0.99

Edge F. 0.79 0.32 0.27 0.99 1

z

Nz

Mz

Nx

xMx

NyMy

y



z

Nz

Mz

Nx

xMx

NyMy

y

Element failure probabilityElement failure probability

i j

Laminate

i j

Assuming :•Laminate is a series system of layers•Each ply has one failure mode described by the specified failure criterion•Positive correlated failure modes among the layers


z

Nz

Mz

Nx

xMx

NyMy

y

Element failure probabilityElement failure probability

i j

Laminate

i j

1

1

max , ,

max , ,

max ,

node i layer layer n

node j layer layer n

element node i node j

f f f

f f f

f f f

P P P

P P P

P P P


z

Nz

Mz

Nx

xMx

NyMy

y

Failure criterionFailure criterion

i j

Laminate

i j

Limit state function for every ply formulated by Tsai-Hahn failure criterion

1

1

max ,

max , ,

max , ,

element node i node j

node i layer layer n

node j layer layer n

f f f

f f f

f f f

P P P

P P P

P P P

0 , 0g fail g safe 1g X R


Reliability methods: Monte CarloReliability methods: Monte Carlo

Basic Variables (RVs)

E1, E2, v12, G12

XT, XC, YT, YC, SNx, Ny, Nz, Mx, My, Mz,

THIN analysis Output Variable (RV)

εx, εy, εs

Limit state function

g x R 1

• Random number generation for the basic variables•THIN analysis•Evaluation of limit state function• Layer failure probability

•Estimation of element failure probability• 2,000,000 simulations

0layer

failsf

total

nP P g n


2

3 4 63 343 322 222 2

22 2

1 2, 2 2, 3,2 21 1 1

1 1 103

4! 6!3!

1, , ...

2

0layer

g

M M M

i i ii i ii i i i

f

F g g g g g

g g g gg

X X X X

P P g

Reliability methods: EDWReliability methods: EDW

Basic Variables (RVs) Output Variable (RV)


g x R 1

E1, E2, v12, G12



Reliability methods: RSM/MCReliability methods: RSM/MC

Basic Variables (RVs) Regression models Output Variable (RV)


g x R 1

• Random number generation for the basic variables•Stress-strain analysis through regression models•Evaluation of limit state function• Layer failure probability

•Estimation of element failure probability• 2,000,000 simulations

0layer

failsf

total

nP P g n

εx, εy, εs (top and bottom layers)

E1, E2, v12, G12



εx, εy, εs (top and bottom layers)

Reliability methods: RSM/MCReliability methods: RSM/MC

Input Variables (RVs)

E1, E2, v12, G12

Nx, Ny, Nz, Mx, My, Mz,

Regression models Output Variable (RV)


g x R 1

Building regression models

Design of Experiment10 input variables5 levels to be tested for every input variable (circumscribed CCD)6 output parameters (strains at the lower face of the bottom and the upper face of the top ply of the laminate).149 THIN analyses


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50

Failu

re p

roba

bilit

y

# element

MC

RSM

EDW

Very good agreement between MC and RSM/MC.EDW is less accurate. (Correlation was not considered)

Failure Probability (IFF): Section 9.2mFailure Probability (IFF): Section 9.2m

[90][90]

[90]

[90][90]

[90]

[90]

[45] [45] [45]

[45]

[45]

[45]

[45]

[-45]

[-45][45]

[45][-45]

[-45]

[-45]

[90][90]

[90][90][90][90][90]

[90]

[90]

[90]

[90][90]

[90]

[90]

[90]

[45]

[-45]

[90]

[45]

Layer #

1

6

1220

26

32

40

41

44

45

48

Element #


Failure Probability (IFF): Section 9.2mFailure Probability (IFF): Section 9.2m

[90][90]

[90]

[90][90]

[90]

[90]

[45] [45] [45]

[45]

[45]

[45]

[45]

[-45]

[-45][45]

[45][-45]

[-45]

[-45]

[90][90]

[90][90][90][90][90]

[90]

[90]

[90]

[90][90]

[90]

[90]

[90]

[45]

[-45]

[90]

[45]

Layer #

1

6

1220

26

32

40

41

44

45

48

Element #

Reliability analysis3h-MC, 30 min-RSM/MC, 10 sec-EDW


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50

Failu

re p

roba

bilit

y

# element

MC

RSM

EDW

ConclusionsConclusions

• Assessment of the reliability level of a rotor blade already designed according to IEC ed. 3 at the ply level was performed

• The stochastic modeling of sectional stress resultants of the blade under extreme loading was achieved

•The probabilistic analysis was performed by using MC, RSM/MC, EDW

• A numerical tool was developed that can be combined with aero elastic codes and can be used for reliability analysis and probabilistic design


ewea 2011 brussels, belgium: europe’s premier wind energy event

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